Visualisatie BMT Algorithms - 2 Arjan Kok a.j.f.kok@tue.nl 1 - - PowerPoint PPT Presentation

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Visualisatie

BMT

Algorithms - 2 Arjan Kok a.j.f.kok@tue.nl

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Lecture overview

• Vector algorithms
• Tensor algorithms
• Modeling algorithms

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Vector algorithms

• Vector
• 2 or 3 dimensional representation of direction and

magnitude (e.g. speed, force)

force = (0, -0.5) force = (2,1) force = (1,1) speed = (3,2)

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Vector algorithms

• Hedgehogs
• Warping
• Oriented glyphs
• Displacement plots
• Time animation
• Stream lines, streak lines
• Stream ribbons, stream surfaces

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Time animation

• Move point or object over a small time step
• Velocity (vector) at position x:

V(x) = dx/dt

• Displacement of point:

dx = V(x)dt

• Repeatedly displace points over many time steps

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Time animation

• Position of point at time t:

∫ =

t

dt )) t ( x ( V ) t ( x

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Time animation

• Give initial position x0

(release point)

• Repeat
• Find cell (i, j, k) and offsets (r, s, t) for x

(point location)

• Find velocity V(x) at x

(interpolation)

• Compute next position

(integration) for given time step Δt

• Until total time passed or new position is outside data set

∫ + = ∆ +

∆ + t t t

dt )) t ( x ( V ) t ( x ) t t ( x

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Time animation

• Integration techniques
• Euler
• x(t+Δt) = x(t) + V(x(t)) Δt
• Runge-Kutta
• x(t+Δt) = x(t) + Δt /2 (V(x(t)) + V(x*(t+Δt)))
• x*(t+Δt) = x(t) + V(x(t)) Δt
• …

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Time animation

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Time animation

• Accuracy depends on:
• Step size Δt
• Accuracy of data set
• Accuracy of interpolation functions

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Particle path

• Particle trace
• Trajectory of a single particle over time
• Path of one particle released at P at t = t0 to t = tn

P t = t0 t = tn

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Streak line

• Streakline
• Path of particles passing through one given fixed point P
• ver a time interval
• Connection of all particles at time ti that have previously

passed through point xi (e.g. source)

P t = t0 t = tn

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Stream line

• Streamline
• Curve in the flow field which at a particular time, in

tangent to the velocity vector at all points along the curve.

• Unsteady flow:
• continuous changing
• given line exists only at one moment in time.

P t = t0 t = tn

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Demo

• http://

widget.ecn.purdue.edu/~meapplet/java/flowvis/Index.html

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Streamlines

• Visualization:
• Define starting points by

creating rake (source of points)

• Draw stream lines
• Color lines according to

some attribute

• Magnitude of vector
• Other scalar attributes

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Streamlines

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Stream ribbon

• Generate two adjacent streamlines
• Generate polygons by connecting point of both lines
• Shows twist (vorticity) and divergence (spread)

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• Generate n streamlines passing starting at points on a curve

(rake)

• Generate polygonal mesh by connecting adjacent

streamlines

• Closed surface => streamtube

Stream surface / stream tube

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Streamtubes

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Tensor algorithms

• A tensor is a high dimensional quantity

(in our case symmetric 3x3 matrices)

• Tensors describe e.g. the displacement and stress in a 3D

material

• We must study the eigenvectors and eigenvalues of the

matrix:

• Ax = λx

where A is tensor matrix and x its eigenvector

• det|A – λI| = 0

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Tensor algorithms

• Express eigenvectors as:
• vi = λiei

where ei is unit vector in direction eigenvalue and λi eigenvalues

• Order eigenvalues such that
• λ1 ≥ λ2 ≥ λ3

(major, medium and minor values)

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Tensor ellipsoids

• Define an ellipsoid
• The shape and orientation of the ellipsoid represent the

relative size of the eigenvalues and the orientation of the eigenvectors

• Algorithm
• Position a sphere at the tensor location
• Rotate the sphere around its origin

(using eigenvectors)

• Scale the sphere (using eigenvalues)

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Tensor ellipsoids

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Tensor ellipsoids

• Point load applied to elastic

material

• At the surface of material the

ellipsiods flatten because there is no stress perpendicular to the surface (except at stress point itself)

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Tensor hyperstreamlines

• Decompose tensor field into three vector fields defined by
• ne of the eigenvectors
• Create a streamline through one of the vector fields
• Sweep an object (e.g. ellipse) along this streamline
• Use remaining vector fields to define major and minor

axis of ellipse

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Tensor hyperstreamlines

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Modeling algorithms

• Modeling algorithms
• Create or change dataset geometry or topology
• Algorithms on combined data
• Most general class of algorithms

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Source objects

• Modeling simple geometry
• Supporting geometry
• Data attribute creation
• Source objects can be used as procedures to create data

attributes (e.g. a simulator)

• Attributes can be generated from a mathematical

function

• => implicit functions

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Implicit functions

• Implicit functions
• F(x, y, z) = c
• Examples:
• Sphere: F(x, y, z) = x2 + y2 + z2 – R2
• Plane:

F(x, y, z) = Ax + By + Cz + D

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Implicit functions

• Properties of implicit functions
• Simple geometric description
• Easy definition to define common geometric shapes

(spheres, planes, cylinders, ..)

• Region separation
• Implicit functions separate 3D space

F(x, y, z) < 0, F(x, y, z) = 0, F(x, y, z) > 0

• Scalar generation
• Implicit functions convert position in space into

scalar value ci = F(xi, yi, zi)

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Use of implicit functions

• Modeling objects
• Sample F on dataset and generate isosurface at contour

value ci

• Combine implicit functions to create more complex
• bjects using boolean operations (union, intersection,

and difference

• F + G = min(F(x, y, z), G(x, y, z))
• F * G = max(F(x, y, z), G(x, y, z))
• F – G = max(F(x, y, z), -G(x, y, z))

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Use of implicit functions

• Modeling example
• f1 = (x-1.33)2 + y2 + z2 – 0.52
• f2 = (x-1.5)2 + y2 + (z-0.5)2 – 0.252
• f3 = f1 – f2
• f4 = y2 + z2 – x2tan2(20) (+ intersection with 2 planes)
• f5 = f4 + f3

f1 f2 f4 f3 f5

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Use of implicit functions

• Selecting data
• Choosing cells and points that lie within a particular

region of the dataset

• Algorithm
• For each cell in dataset
• Evaluate implicit function for all cell nodes
• A cell is selected if result has negative sign

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Use of implicit data

• Data selection

Selecting with sphere function

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Use of implicit functions

• Data cutting
• Find dataset values on implicit surface

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Use of implicit functions

• Data cutting
• Algorithm
• For each cell in dataset
• Evaluate implicit function for all cell nodes
• A cell is cut if not all positive or negative
• Generate the isosurface f(x, y, z) =0 within cell
• Generate data attributes for isosurface by

interpolation along cut edges

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Use of implicit functions

• Data cutting

Data cutting with plane function

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Use of implicit functions

• Clipping
• Limit the data to be processed or displayed
• Usually with plane

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Use of implicit functions

• Clipping
• Algorithm
• Evaluate implicit function for all cell nodes
• If all positive => remove cells
• If all negative => copy cells
• Else
• Generate the isosurfaces f(x, y, z) =0 within cell

and generate new cells.

• Generate data attributes for isosurface by

interpolation along cut edges

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Glyphs

• A glyph is any object which is parameterized by some data
• May be positioned, oriented, scaled, deformed, colored,

.. in response to data

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Data extraction

• Data extraction
• Extract portions of data from dataset
• Techniques:
• Geometry extraction
• Thresholding

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Data extraction

• Geometry extraction
• Extracts data based on geometric or topological

characteristics

• Select cells/nodes within specified range of ids
• Spatial extraction
• Define regions (e.g. by implicit function)
• See data selection
• Subsampling
• Select part of original data
• Only select every n-th data node
• Modifies topology of dataset

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Data extraction

• Thresholding
• Extracts data based on attribute values
• Examples
• Select all nodes with scalar values within specified

range

• Select all nodes with vector magnitude within

specified range

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Probing

• Resampling method
• Create output dataset by sampling input dataset
• Get attributes for output dataset by interpolation from

attributes of input dataset

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Probing